Key points are not available for this paper at this time.
In this paper, we consider two types of semi-online problems with hierarchies. In the extensible bin-packing problem with two hierarchical bins, one bin can pack all items, while the other bin can only pack some items. The initial size of the bin can be expanded, and the goal is to minimize the total size of the two bins. When the largest item size is given in advance, we provide some lower bounds and propose online algorithms. When the total item size is given in advance, we provide some lower bounds and propose online algorithms. In addition, we also consider the relevant early-work-maximization problem on two hierarchical machines; one machine can process any job, while the other machine can only process some jobs. Each job shares a common due date, and the goal is to maximize the total early work. When the largest job size is known, we provide some lower bounds and propose two online algorithms whose competitive ratios are close to the lower bounds.
Yang et al. (Mon,) studied this question.